Appendix A Powers of Ten

Conclusion This has been a theory book for observational amateur astronomers. This is perhaps a bit unusual because most astronomy theory books tend to be written for armchair astronomers and they tend to deal with fairly exotic subjects like black holes and cosmology. Spectroscopy however is different; for one thing it s a fairly new area of research for amateurs, so until now there hasn t been much of a compelling need for a book like this. The other thing is that spectroscopy by its very nature involves a lot of physics and most amateur astronomers don t have physics degrees. There was, I reckon, an urgent need to provide an explanation of some of the theory behind spectroscopy, including the relevant physics. That was the main idea behind this book. I hope that by now you ve gained a deeper insight into spectroscopy and most of all come to realise (if you didn t realise already) that spectroscopy involves far more than just identifying lines in spectra. Indeed, identifying spectral lines is very likely the subject for another book; one which needs to be written by someone with many years of observational experience. Maybe out there, there s a kind hearted professional astronomer or a pioneering amateur who could do this. If, as a result of reading this book however, you can better appreciate the things which are going on in your spectra, thenireckoni vedonemyjob. Clear skies and good luck with your spectra. 149

Appendix A Powers of Ten First take as an example the number 2.9 and multiply it by 10: we get 29 of course 2.9 10 = 29 Similarly 2.9 100 = 290 2.9 1000 = 2900 and so on. We are simply moving the decimal point one place to the right each time. Now take 2.9 100,000,000,000 Here we have to make a conscious effort to count the zeros and for each one of them, move the decimal point one place to the right. So we get 290,000,000,000 This is not a very elegant way to write two hundred and ninety thousand million. A much better way is to use scientific notation. 100 = 10 10, i.e. 10 multiplied by itself twice 1000 = 10 10 10, i.e. 10 multiplied by itself 3 times 10,000 = 10 10 10 10, i.e. 10 multiplied by itself 4 times and so on. Scientific notation writes these numbers like this 100 = 10 2 1000 = 10 3 10000 = 10 4 and so on. 151

152 Spectroscopy The Key to the Stars You can see the advantage of writing a number like 100,000,000,000, i.e. one hundred thousand million as 10 11. You pronounce this ten to the eleven A number like 10 11 is also referred to as 10 to the power 11 or 10 raised to the power 11. Back to our number 2.9; now instead of writing 2.9 100,000,000,000, we can simply write this as 2.9 10 11 This is an example of scientific notation. The 10 raised to the power bit is called the exponent. Let s take our 2.9 10 11 It comes in two parts: 2.9 and 10 11 We could multiply the 2.9 by 10 to get 29 and then we would have 10 2.9 10 11 = 29 10 11 However, you would never write this number like this. The first part of a scientific notation number should always be greater than 1 and less than 10. We keep the first bit as 2.9 and so we multiply the 10 11 bit by 10 instead. 10 11 10 becomes 10 multiplied by itself 12 times, i.e. 10 12. So our scientific notation number is written as 2.9 10 12 The 11 in 10 11 is often referred to as the index of the power of 10. Notice above that when we multiplied 10 11 by 10 we simply increased the value of the index by 1 to get 10 12. Had we multiplied 10 11 by 100 (i.e. 10 2 ) we would have got 10 13. Multiplying by 1000 (10 3 ) would have given us 10 14 and so on. In other words, when we multiply two powers of 10 together, we simply add the two indices; e.g. 10 7 10 11 = 10 18 etc. Just as 10 2 is the number 1 with 2 zeros after it and 10 3 is the number 1 with 3 zeros after it etc. the number 10 on its own is the number 1 with 1 zero after it, i.e. 10 = 10 1 ; and the number 1 is of course 1 with no zeros after it, i.e. 1 = 10 0. Now let s multiply together, two numbers in scientific notation, so let s work out 2.9 10 7 3.6 10 5 First, multiply the powers of 10 Using our calculator 10 7 10 5 = 10 12 2.9 3.6 = 10.44 So this would initially give us 10.44 10 12 but remember we keep the first part of the number less than 10. So we divide it by 10 to give us 1.044 and we must multiply the exponent (the powers of 10 bit) by 10 to balance the books. So we get 1.044 10 13

Appendix A Powers of Ten 153 Now let s try dividing two powers of 10; let s try 10 5 /10 2 This of course is just 100,000/100 the zeros cancel out to give 1000. 1000 is of course 10 3, but notice that 3 = 5 2. In other words, when we divide one power of 10 by another, we simply subtract the lower power from the upper power, e.g. 10 12 /10 7 = 10 5 So far we ve talked about a power of 10 in terms of the number of zeros after the 1 by which we mean of course the number of zeros to the right of the 1 and we get a positive index for our power of 10. So if we think of a zero to the right of the 1 as contributing to a positive index for the power of 10, we could think of a zero to the left of the 1 as contributing to a negative power of 10 index. Thus the number 0.1 (written this way as opposed to.1) has one zero to the left of the 1 and in fact 0.1 = 1/10 = 10 1 0.01 = 1/100 = 1/10 2 = 10 2 : the 1 has 2 zeros to its left 0.001 = 1/1000 = 1/10 3 = 10 3 : the 1 has 3 zeros to its left So for example 0.000001 = 1/1000000 = 1/10 6 = 10 6 This is consistent with the rules for multiplying and dividing powers of 10, e.g. 10 7 10 4 = 10 7 1/10 4 = 10 7 /10 4 = 10 (7 4) = 10 3 Summarising so far, we can use scientific notation with a positive power of ten index to very conveniently represent large numbers; e.g. 2.9 10 18 and by using a negative power of ten index, we can neatly represent very small numbers, e.g. 2.9 10 18. What about square roots and cube roots and so on? Can we use scientific notation here? 100 is 10 10 which is 10 2 of course. 10 is the square root of 100. But 10 is of course 10 1. So to get the square root of 100 we divided the index on 10 2 by 2; i.e 10 2 is simply 10 2/2 which equals 10 1, i.e. 10. Thesquarerootof10 4 is 100, i.e. 10 2. Once again to get the square root, we divide the power of ten index by 2. 10 is the same as 10 1 and so by the same token, to get the square root, we divide the power of ten index by 2. So 10 = 10 1 = 10 1/2 So 10 = 10 1/2 In a similar way the cube root of 10 is given by 10 1/3 ; the fourth root by 10 1/4 etc. Now take for example the number (10 4 ) 6 ; this is not the same as 10 4 10 6,itisthe number 10 4 multiplied by itself 6 times. Do this longhand and you can see that you geta1with24zerosafterit;i.e. (10 4 ) 6 = 10 (4 6) = 10 24 Now note that a number like (2.9 10 4 ) 6 is equal to (2.9) 6 10 24 Right! Let s do a full-blown formula in scientific notation. Back in the 1970s, theoretical physicists were thinking about an interval of space which was so small that Einstein s general theory of relativity breaks down and is replaced by a still unknown quantum theory of gravity. How big would this interval be? The formula they arrived at is given

154 Spectroscopy The Key to the Stars by [ Gh L = L is called the Planck length, G is the universal constant of gravitation = 6.67 10 11, h is Planck s constant = 6.6 10 34, c is the speed of light = 3 10 8 So we have [ 6.67 10 11 6.6 10 34 ] 1/2 L = (3 10 8 ) 3 Let s sort out the powers of 10 first 10 11 10 34 gives us 10 45 (10 8 ) 3 = 10 24 10 45 /10 24 = 10 69 Now the number bits 6.67 6.6/3 3 = 44.022/9 = 4.89 So far we have 4.89 10 69 but we need the square root of this. To get the square root of 10 69 we need to divide the index of the exponent by 2 but this is awkward unless we have an even numbered index. So, breaking the rule above (we are after all in the middle of a calculation and we ll be sure to write the number correctly at the end) let s write 4.89 10 69 as 40.89 10 70, i.e. we ve multiplied the number bit by 10 and divided the power of 10 by 10. Thesquarerootof10 70 is 10 35 and the square root of 40.89 is (use calculator) 6.39. So our final answer in correct scientific notation is L = 6.39 10 35 and this distance is in metres by the way a very tiny distance indeed. c 3 ] 1/2

Appendix B Constants and Formulae I ve gathered together here some of the more important physical constants and useful plug the numbers in formulae which I ve used in the book. As well as enabling you to easily calculate useful numbers, these formulae are the key to exploring some piece of spectroscopic theory and getting a real feel for what s going on. Physical Constants These are in what s known as the MKS or metre (m) kilogram (kg) second (s) system; also known as the SI system. Speed of light c: 2.998 10 8 m/s. Planck s constant h: 6.626 10 34 Js. Gravitational constant G: 6.673 10 11 Nm 2 /kg 2 Boltzmann constant k: 1.381 10 23 J/K Mass of hydrogen M H :1.674 10 27 kg One electronvolt 1 ev: 1.602 10 19 J One angstrom 1 Å: 10 10 m 155

156 Spectroscopy The Key to the Stars Astronomical Constants One Solar mass: 1.99 10 30 kg One (sidereal) year: 365.256 (mean solar) days = 31,558,118 s. One astronomical unit (AU): 1.496 10 11 m Formulae Energy E Equivalent to Wavelength λ This gives the wavelength in angstroms which is equivalent to energy in electronvolts. Wavelength in angstroms = 1.24033 10 4 /energy in electronvolt (see Chapter 3). Doppler Formula Wavelength change λ in terms of velocity v λ = λ v/c Velocity in terms of wavelength change v = c λ/λ λ and λ are in angstroms; v must be in the same units as c; e.g. metres per second. or kilometer per second. Relativistic Doppler Formula λ = λ 1 + v c 1 + v 2 There s probably little need to use this for velocities less than about 5000 km/s unless you re doing high-resolution spectroscopy (see Chapter 4). c 2 1 Kepler s Third Law Formula Orbital period P (e.g. of a binary) in terms of binary separation a and stars masses M 1 and M 2 ; P 2 = M 1 + M 2 P is in (Earth) years, a in astronomical units (AU) and M 1 and M 2 are in solar masses (see Chapter 4). a 3

Appendix B Constants and Formulae 157 Full Width Half-Maximum (FWHM) Formula The FWHM is the total width in angstroms of a Doppler (or thermally) broadened spectral line at half its maximum intensity (for an emission line) or depth (for an absorption line). The temperature of the gas in Kelvin is given by T = 1.968 10 12 (FWHM) 2 /λ 2 0 λ 0 is the rest wavelength in angstroms of the spectral line (see Chapter 5). Wien s Displacement Law This gives the wavelength λ max in angstroms at maximum emission for a black body (a star is a reasonable approximation to this) of temperature T Kelvin. See Chapter 8 λ max = 28, 978, 200/T Zeeman Effect Formula This gives the separation λ in Angstroms between each of the three components of a spectral line of wavelength λ 0 which is split by a magnetic field of strength H gauss (see Chapter 11) λ = 4.67 10 13 λ 2 0 H

Index absolute zero 15, 58 absorptivity 15 alkali metals 37, 38 angstrom, defined 9 angular momentum, defined 129 quantum number 31 33, 38 40, 130 anomalous Zeeman Effect 139, 140 astronomical unit 49, 156 atomic number, defined 20 Balmer continuum 71 decrement 98, 99, 114 discontinuity 71 jump 71, 77, 80 series 35 37, 43, 70, 77, 114, 144, 145 Boltzmann constant 59 equation 73, 74, 146, 147, 155 bound bound transition 27, 28, 30, 34 36, 42, 43, 93 bound-free transition 27, 29, 72 case A recombination 98 B recombination 98 collision broadening 61, 143 collisional excitation, see excitation, collisional coordinate frame 130 132, 136 colour excess 80 colour index 80, 81 degenerate (levels) 32, 33, 37 39 dispersion 10, 106 Doppler broadening 59, 60, 63 velocity 59 61 doublet 38, 41 43, 77 effective temperature 75, 81, 94 Einstein probability coefficients, see probability coefficients electronvolt, defined 25 emissivity 15 ev, see electron volt excitation, defined 29, collisional 101 103, thermal 29, 43, 73, 93 forbidden radiation 34, 100 forbidden transition 34, 42, 100 103 free-bound transition 27 free-free transition 27, 72 frequency, defined 8 fill width half maximum 60, 63, 157 FWHM, see full width half maximum gauss (unit) 127,128, 132, 138, 139, 157 Gaussian distribution 58 Grotrian diagram 37 ground state 29, 35, 54, 72, 87, 93, 94, 97, 100 h, see Planck s constant Heisenberg uncertainty Principle 55, 56 Inclination (of accretion disk) 108, 110 113, 115 inner quantum number 39, 40, 102 instrumental profile 62 intercombination lines 41 ionisation, defined 27 photo, defined 29 potential, defined 30 thermal, defined 29 isotopes, defined 20 j, see inner quantum number joule, defined 25 Kelvin (temperature scale) defined 15 Kirchoff s laws 13 l, see angular momentum quantum number Larmor precession 134 line strength 144 L-S coupling 39, 41 Lyman series 35, 36, 54, 97 m, see magnetic quantum number magnetic quantum number 31, 32, 128, 133, 140 metastable level(s) 34, 55, 101 103 m l, see magnetic quantum number multiplets 42 n, see principal quantum number nanometre, defined 9 natural line broadening 55, 57 59, 61 63, 68, 78 neutral atom(s) 20, 29, 36, 41, 73, 74, 145, 146 Oscillator strength 144 147 159

160 Index parameter space 106 Paschen Bach effect 139, 140 Paschen continuum 71 series 35, 36, 71 Pauli exclusion principle 24, 42 permitted radiation 34 Planck s constant 17, 18, 21, 28, 154, 155 pressure broadening, see collision broadening principal quantum number(s) 31 33, 37 39 probability coefficients 101 radial motion, see radial velocity radial velocity 45 50, 57 59, 62, 106, 108 117 radiation damping, see natural line broadening radiation field 56, 59, 87, 97, 101, 102, 143 radical(s) 84 reference frame, see coordinate frame relativistic Doppler shift formula 47 Russell Saunders coupling, see L-S coupling spin quantum number 31 34, 38, 39 spin up 33, 34, 37 39, 41, 101 down 33, 37, 38, 40, 41, 101 term diagram(s) 37, 38 tesla 127, 128 thermal broadening, see Doppler broadening thermal excitation, see excitation, thermal transition probability, see probability coefficients transition(s), forbidden 34, 42, 100 103 permitted 34, 42, 101 transition series 35 37, 57, 91, 145, 147 transverse wave 7, 8 triplet(s) 40, 41, 43, 76, 101, 102, 139 turbulence velocity 60 Uncertainty principle, see Heisenberg uncertainty principle s, see spin quantum number Saha s equation 73, 74, 146 saturated (line) 53, 142 144, 146, 147 scattering 28 selection rules 34, 38, 42, 101 shear velocity 115, 118 singlet(s) 41, 42, 76, 101, 102, 139 spectral series, see transition series valence electron(s) 36, 38 41, 77, 85, 95 vibration quantum number 89 Voigt profile 63, 142 wavelength, defined 7 wavenumber, defined 10 Wien s displacement law 94, 157